Question

2. The 2p and 2py wave functions are constructed as linear combinations of the n-2, 1-1, m,- 1 wave functions which are eigenfunctions of the hydrogen atom Hamiltonian. Are 2px and 2py wave functions also eigenfunctions of the hydrogen atom Hamiltonian? In other words, do 2px and 2py wave functions denote the same energy states as n-2, -1, mF+ 1 wave functions of the hydrogen atom? (20 points).

Answer 1

The wave functions Psi_211 = 1/8 squareroot pi a^3 (r/a) e^-r/2a sin theta e^i phi Psi_21 - 1 = 1/squareroot pi a^3 (r/a) e^-r/2a sin theta e^i phi Psi_2 P_x = 1/squareroot 2 (Psi_211 + Psi_21 - 1) sin theta cos phi Psi_2 P_y = i/squareroot 2 (Psi_211 - Psi_21 - 1) sin theta sin phi we have H Psi_211 = elementof_0/2^2 Psi_211 where elementof_0 = -13.6 RV H Psi_211 - 1 = elementof_0/2^2 Psi_21 - 1 Rightarrow H Psi_2P_x = 1/squareroot 2 (H Psi_211 + H Psi_21 - 1) = 1/squareroot 2 (elementof_0/4 Psi_211 + elementof_0/4 Psi_21 - 1) = elementof_0/4 (1/squareroot 2 (Psi_211 + Psi_21 - 1)) = elementof_0/4 Psi_2 P_x \r\n Hence Psi_2p_x is an eigen value of hydrogen Hamiltonian with same energy elementof_0/4 as like Psi_211 Psi_21 - 1 H Psi_2p_y = 1/squareroot 2 (H Psi_211 - H Psi_21 - 1) i/squareroot 2 (elementof_0/4 Psi_211 - elementof_0/4 Psi_21 - 1) = elementof_0/4 i/squareroot 2 (Psi_211 - Psi_21 - 1) = elementof_0/4 Psi_2p_y hence Psi_2p_y is also an eigenstate of hydrogen Hemitonian